# Universality for random tensors

• Volume: 50, Issue: 4, page 1474-1525
• ISSN: 0246-0203

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## Abstract

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We prove two universality results for random tensors of arbitrary rank $D$. We first prove that a random tensor whose entries are ${N}^{D}$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large $N$ limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large $N$ Gaussian isnot universal, but depends strongly on the details of the joint distribution.

## How to cite

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Gurau, Razvan. "Universality for random tensors." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1474-1525. <http://eudml.org/doc/272101>.

@article{Gurau2014,
abstract = {We prove two universality results for random tensors of arbitrary rank $D$. We first prove that a random tensor whose entries are $N^\{D\}$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large $N$ limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large $N$ Gaussian isnot universal, but depends strongly on the details of the joint distribution.},
author = {Gurau, Razvan},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random tensors; large $N$ limit; random tensors; large limit},
language = {eng},
number = {4},
pages = {1474-1525},
publisher = {Gauthier-Villars},
title = {Universality for random tensors},
url = {http://eudml.org/doc/272101},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Gurau, Razvan
TI - Universality for random tensors
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1474
EP - 1525
AB - We prove two universality results for random tensors of arbitrary rank $D$. We first prove that a random tensor whose entries are $N^{D}$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large $N$ limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large $N$ Gaussian isnot universal, but depends strongly on the details of the joint distribution.
LA - eng
KW - random tensors; large $N$ limit; random tensors; large limit
UR - http://eudml.org/doc/272101
ER -

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